let the number= x
the other =12-x
the product be x
x= x(12-x)
x=12x - x^2
dy/dx=12 - 2x
at maximum height dt/dx=0
12 - 2x=0
-2x=-12
x=-12/-2
x=6
d^2y/dx^2= -2< 0
since x=6 the maximum value of y
=6(12-6)
y=36
Let x be the first number. The sum of the two numbers is 12 so 12-x is an expression for the second number.
.
Let y be the product of the two numbers.
.
Then an equation that represents this relationship is
y=x%2812-x%29
.
Expand the right hand side of the equation.
y=12x-x%5E2
.
Re-order the terms.
y=-x%5E2%2B12x
.
This is a quadratic equation; its graph is a parabola. Since the leading coefficient is negative the vertex represents the maximum value of the equation.
.
If the vertex of a parabola is the point (h,k), then h=-b/2a. In the formula, a is the coefficient of the x-squared terms, and b is the coefficient of the x-term when the quadratic equation is in general form [y=ax^2+bx+c].
.
In our equation, a is -1 and b is 12.
h=-b%2F2a
h=%28-12%29%2F%282%2A%28-1%29%29
h=%28-12%29%2F%28-2%29
h=6
.
The equation h=6 means that the x-coordinate of the vertex is 6. Substitute 6 for x in the quadratic equation to find the y-coordinate.
.
y=-x%5E2%2B12x
y=-%286%29%5E2%2B12%286%29
y=-36%2B72
y=36
.
The y-coordinate of the vertex is 36, so the vertex is (6,36).
.
Now we need to interpret our answer in terms of this problem. Recall that x is the first number. The first number is 6. The second number is also 6 since 12-x=12-6=6. The y-coordinate is the product of the two numbers, 36.
3 months ago
the other =12-x
the product be x
x= x(12-x)
x=12x - x^2
dy/dx=12 - 2x
at maximum height dt/dx=0
12 - 2x=0
-2x=-12
x=-12/-2
x=6
d^2y/dx^2= -2< 0
since x=6 the maximum value of y
=6(12-6)
y=36